Cool powergyoza, that silently sounds like a pretty great reduction . How did you put them together? We can conclude that the rubber idea works well, and if we need to improve it it's only the thermal issue that might want some help.
you might be able to get a better thermal interface by just putting a thin layer of caulking, maybe ¼" thick, on the bottom and top layers of rubber, then stick them onto the drive. Fill them air gaps! As long as warranty isn't an issue, I would just bond them right on, since there's nothing there you need access to. Same idea aplies to the sides. Alternatively you could try doing a seran wrap conformal layer type trick to give the top and bottom a closer form fitting profile without the pemanence.
After trying to probe my brain about types of caulking to use, I think Mono might be good because it is solvent based instead of acetic acid like true silicone, or water like the latex stuff, so I think it would be less likely to harm the drive. Probably the very best, but more costly, would be a uerethane like they black stuff used for automotive windsheild seams, or marine Sikaflex, both of which will never
come off, but cure by absorbing moisture from the surrounding air, instead of drying. Better living through chemistry eh?
I know it's a little late for you now, but conveyor belt rubber can be got used, or probably bought new as small scraps for peanuts, also think truck tire flap rubber. I don't know how the fibers would affect the thermal conductivity though, probably not good.